The Watched Over Zone

In general, for a $m\times n$ chessboard, with $m\le n$ , and $k$ Rooks, with $k\le m$ , as each Rook attacks $m+n-2$ squares (its whole column and row, and we're counting twice the square on which it is stood) and two Rooks attack the same $2$ squares (the intersection of the rows and columns they're attacking), the number of squares under attack is given by the following expression:

$k(m+n-2)-2\left( \begin{matrix} k \\ 2 \end{matrix} \right)$

In the particular case $m=n=100$ and $k=50$ , the number of squares under attack is $50(100+100-2)-2\left( \begin{matrix} 50 \\ 2 \end{matrix} \right) =50\cdot 198-50\cdot 49=7450$ .